Families of explicit quasi-hyperbolic and hyperbolic surfaces

TL;DR

This paper constructs explicit families of hyperbolic and quasi-hyperbolic surfaces, extending previous work to include singular cases and providing new examples of surfaces of general type with hyperbolic properties.

Contribution

It extends the construction of hyperbolic surfaces to singular cases and produces explicit examples of smooth complete intersection surfaces of various degrees and dimensions that are hyperbolic.

Findings

Explicit families of hyperbolic surfaces in high dimensions.

Extension of hyperbolic surface constructions to singular cases.

Evidence supporting conjectures on hyperbolic surfaces of general type.

Abstract

We construct explicit families of quasi-hyperbolic and hyperbolic surfaces. This is based on earlier work of Vojta, and the recent expansion and generalization of it by the first author. In this paper we further extend it to the singular case, obtaining results for the surface of cuboids, the generalized surfaces of cuboids, and other families of Diophantine surfaces of general type. In particular, we produce explicit families of smooth complete intersection surfaces of multidegrees $(m_1,\ldots,m_n)$ in $\mathbb{P}^{n+2}$ which are hyperbolic, for any $n \geq 8$ and any degrees $m_i \geq 2$. We also show similar results for complete intersection surfaces in $\mathbb{P}^{n+2}$ for $n=4,5,6,7$. These families give evidence for [Dem18, Conjecture 0.18] in the case of surfaces.